The Loewner framework for model reduction of large-scale systems: An Overview and Sensitivity considerations Thanos Antoulas Rice University, Houston, and Max-Planck Institute, Magdeburg e-mail: aca@rice.edu URL: www.ece.rice.edu/ ˜ aca Mathematics of Reduced Order Models The Institute for Computational and Experimental Research in Mathematics ICERM: 17-21 February 2020 1 / 49
Introduction ”Model order reduction (MOR) is a methodology for reducing the computational complexity of mathematical models in numerical simulations. [Wikipedia] Here (1) we will give an overview of the data-driven modeling framework known as the Loewner framework and (2) we will discuss issues related to the sensitivity of the resulting models . The overall problem Physical or artificial system ⇓ PDEs ⇓ ODEs ⇓ = DATA Model reduction ⇐ (Reduced number of ODEs) ⇓ Simulation, Design, Control 2 / 49
Dynamical systems u 1 ( · ) − → − → y 1 ( · ) � ˙ x ( t ) = f ( x ( t ) , u ( t )) u 2 ( · ) − → − → y 2 ( · ) Σ : y ( t ) = h ( x ( t ) , u ( t )) . . . . . . u m ( · ) − → − → y p ( · ) We consider internal equations E ˙ Σ : x ( t ) = f ( x ( t ) , u ( t )) , y ( t ) = h ( x ( t ) , u ( t )) with internal variable x ( · ) of dimension n > m , p . 3 / 49
Model reduction via projection E ˙ x ( t ) = f ( x ( t ) , u ( t )) E ˙ x ( t ) = Ax ( t ) + Bu ( t ) Given is or . y ( t ) = h ( x ( t ) , u ( t )) y ( t ) = Cx ( t ) + Du ( t ) Common framework for (most) model reduction methods: Petrov-Galerkin projective approximation . k ) ⊂ C n . Choose k -dimensional subspaces, V k = Range ( V k ) , W k = Range ( W Find v ( t ) = V k x k ( t ) ∈ V k , x k ∈ C k , such that E ˙ v ( t ) − Av ( t ) − B u ( t ) ⊥ W k ⇒ W ∗ k ( EV k ˙ x k ( t ) − AV k x k ( t ) − B u ( t )) = 0 , y k ( t ) = CV k x k ( t ) + Du ( t ) , Reduced order system E k = W ∗ A k = W ∗ B k = W ∗ k EV k , k AV k , k B , C k = CV k . 4 / 49
The quality of the reduced system depends on the choice of V k and W k . n k n E , A B k E k , A k B k ⇒ C D C k D k Model order reduction (MOR) seeks to reduce the complexity of large-scale dynamical systems by approximations of much lower dimension that produce nearly the same input/output response characteristics. In the sequel we will concentrate on interpolatory model reduction methods and in particular on the Loewner framework which is data-driven . 5 / 49
Motivating Examples ◮ Model reduction of an artificial fishtail . • The autonomous soft robotic fish was developed at MIT and Worchester Polytechnic. The fish is novel because it is has a soft body and has a new fluidic actuation system. soft- : A fish anterior, an- actu- inextensible (E) pair, fin. cross-sectional mechanism elasto- into (H) and elastomer actua- exploded detailing com- elec- gas ) ac- plastic videography silicone A.D. Marchese, C.D. Onal, and D. Rus, Autonomous soft robotic fish capable of escape maneuvers using fluidic elastomer actuators , Soft Robotics, vol. 1, pages 75-87, (2014). • Mathematical model . Constitutive equations (elasticity): ρ ∂ 2 t s ( t , z ) = ∇ · σ ( s ( t , z )) , t > 0 , z ∈ Ω , where s ( 0 , z ) = s 0 ( z ) , where s ( t , z ) is the first-order displacement tensor, σ ( t , z ) is the stress tensor and Ω is the domain of interest. 6 / 49
By appropriate discretization of this partial differential equation, we obtain a second-order finite-dimensional system of the form M ¨ x ( t ) + E ˙ x ( t ) + K x ( t ) = Bu ( t ) , y ( t ) = Cx ( t ) , where x is a discretized version of s , and the damping E is proportional. Goal: build a controller, for the tail. The main issue at this stage is the resulting complexity, namely: M , E , K ∈ R N × N , B ∈ R N , C ∈ R 3 × N , N = 779 , 232 . where D. Siebelts, A. Kater, and T. Meurer, Modeling and motion planning for an artificial fishtail , IFAC PapersOnLine, vol. 51, pages 319-324 (2018). 7 / 49
• Model order reduction J. Saak, D. Siebelts, and S.W.R. Werner, A comparison of second-order model order reduction methods for an artificial fishtail , Automatisierungstechnik, vol. 67, pages 648-667 (2019). Procedure . Choose n = 500 frequencies on the imaginary axis: omega = logspace ( − 1 , 3 , 500 ) . Subsequently compute the 500 frequency response values: � � − 1 s 2 M + s E + K B ∈ C 3 × 1 , where s = j ω k , k = 1 , · · · , 500 . H ( s ) = C Remark : this computation takes about 2 days. Together with the complex conjugate frequencies and transfer function values build the Loewnere quadruple: L , L s ∈ R n × n , V ∈ R n , W ∈ R 3 × n . 8 / 49
◮ Accelerating the commercial aircraft design and certification cycle . • The design and optimization of modern commercial aircraft typically requires accurate predictions of airframe response to diverse operational loads, which may include for example, wind gusts, turbulence, wake vortices, asymmetric thrust, and vibration. • This must be done repeatedly at different stages of the design process. • High fidelity reduced models play a significant role in reducing overall simulation time by providing cheap and accurate surrogates for different subsystems. • The data-driven Loewner approach is well suited to this purpose since the construction of reduced models can be done independently of the availability of analytical system models. • The system input-output response was measured at 421 frequencies. • The system input is a distributed gust disturbance. • The system output is the lift and pitch moment at 44 locations on the wing and tail. • The full-order system has 5 × 10 5 fluid degrees of freedom and 2 × 10 3 structural degrees of freedom. • The reduced dynamical system has 30 degrees of freedom. • The relative deviation between the reduced system response and the full-order system response is never larger than 0 . 35 % 9 / 49
10 2����� � � � �-�-�-�����- - - - 1� 10 1 10 ° � �-�-�-�����- - - - -����- - - - � � - � ����- - - � - - 10-7 �� � 10 -6 ����- - - - �� � ���������� � � � ������������� � - � 10 -6 10 2����- - - �- - �-�� Original (full-order) frequency response Reduced (order 30) frequency response Matlab Loewner Toolbox Dassault business jet test C. Poussot-Vassal & P. Vuillemin 10 / 49
◮ Microstrip device . Microstrip transmission lines are a common way to connect two devices. They consist of two thin parallel (metal) strips, mounted on the same dielectric substrate. Microstip devices are described by S-parameters (Scattering parameters). VNA (Vector Network Analyzer) and VNA screen showing the magnitude of the S -parameters for a 2 port device. 11 / 49
The Loewner matrix Karel L¨ owner (1893 - 1968) Given: ( µ j , v j ) , j = 1 , · · · , q , row array column array ( λ i , w i ) , i = 1 , · · · , k , the associated Loewner matrix is: v 1 − w 1 v 1 − w k · · · µ 1 − λ 1 µ 1 − λ k . . ... ∈ C q × k L = . . . . • Born in Bohemia v q − w 1 v q − w k · · · µ q − λ 1 µ q − λ k • Studied in Prague under Georg Pick • Emigrated to the US in 1939 If w i = g ( λ i ) , v j = g ( µ j ) , are samples of g : • Seminal paper: Main property . Let L be as above. ¨ Uber monotone Matrixfunctionen, Then k , q ≥ deg g ⇒ rank L = deg g . Math. Zeitschrift (1934). 12 / 49
Rational interpolation and the Loewner matrix • Lagrange basis for space of polynomials of degree at most n . Given distinct λ i ∈ C , i = 1 , · · · , n + 1, define q i ( s ) := Π k � = i ( s − λ k ) , i = 1 , · · · , n + 1 . • For given constants α i , w i , i = 1 , · · · , n + 1, consider n + 1 � g − w i = 0 , α i � = 0 . α i s − λ i i = 1 • Solving for g we obtain α i w i � n + 1 s − λ i i = 1 ⇒ g ( s ) = g ( λ i ) = w i . � n + 1 α i i = 1 s − λ i 13 / 49
The free parameters α i , can be specified so that g ( µ j ) = v j , j = 1 , · · · , r , where ( µ j , v j ) , µ i � = µ j , are given. For this to hold L c = 0, where v 1 − w n + 1 v 1 − w 1 · · · α 1 , µ 1 − λ 1 µ 1 − λ n + 1 . . . ... ∈ C r × ( n + 1 ) , c = ∈ C n + 1 . L = . . . . . . v r − w n + 1 v r − w 1 α n + 1 · · · µ r − λ 1 µ r − λ n + 1 L : Loewner matrix with row array : ( µ j , v j ) , j = 1 , · · · , r , and i = 1 , · · · , n + 1. column array : ( λ i , w i ) , Main property Let L be a p × k Loewner matrix with p , q ≥ deg g . Then rank L = deg g . • A.C. Antoulas and B.D.O. Anderson, On the scalar rational interpolation problem , IMA Journal of Mathematical Control & Information, vol. 3, pages 61-88 (1986). 14 / 49
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